EDIT: after listening to the podcast linked by OP, Mr. Shapiro never says the words claimed by OP, and never says that the subpoenas are illegitimate. It is possible that he linked to the wrong podcast, or it was a comment on the video itself. I've made edits to my answer to reflect this information.
Congress has the power to issue subpoenas and has issued multiple subpoenas to members of the Trump Executive Branch, as well as his personal lawyer.
Failure to comply with a subpoena is considered Contempt of Congress.
Contempt of Congress is the act of obstructing the work of the United States Congress or one of its committees. Historically, the bribery of a U.S. Senator or U.S. Representative was considered contempt of Congress. In modern times, contempt of Congress has generally applied to the refusal to comply with a subpoena issued by a Congressional committee or subcommittee—usually seeking to compel either testimony or the production of requested documents.
Both houses of Congress have the ability to issue subpoenas, and all standing committees are allowed to also issue subpoenas.
Congressional rules empower all its standing committees with the authority to compel witnesses to produce testimony and documents for subjects under its jurisdiction. Committee rules may provide for the full committee to issue a subpoena, or permit subcommittees or the chairman (acting alone or with the ranking member) to issue subpoenas.
A list of standing committees can be found here. Of note, the following committees are considered standing committees.
By contrast, the United States House Permanent Select Committee on Intelligence is not considered a standing committee, but is instead a "permanent select" committee. However, it appears that the Intelligence Committee is allowed to issue subpoenas, as former chairman Rep. Devin Nunes issued multiple subpoenas in 2017 and 2018 while he was chairman.
From CNN, in 2017
The anger behind the scenes was reignited when Nunes signed off on four subpoenas sent by House Russia investigators to former national security adviser Michael Flynn and Trump lawyer Michael Cohen -- and also issued three subpoenas on his own, seeking information from former Obama administration officials.
From lawfareblog.com, regarding 2018
In April of 2018, Nunes threatened to hold FBI Director Christopher Wray and Deputy Attorney General Rod Rosenstein in contempt and start impeachment proceedings because for more than seven months, Wray and Rosenstein had failed to fulfill Nunes’s request for an unredacted copy of a two-page memo the FBI used to initiate its investigation of the Trump campaign’s Russia contacts.
From USA Today on 9 October 2019, here is a list of all subpoenas issued with regards to the Ukraine investigation.
- European Union Ambassador Gordon Sondland (Oct. 8)
- Defense Secretary Mark Esper/The Pentagon (Oct. 7)
- Acting Office of Management and Budget Director Russell Vought (Oct. 7)
- Acting White House Chief of Staff Mick Mulvaney/The White House (Oct. 4)
- Vice President Mike Pence (Oct. 4)
- Rudy Giuliani (Sept. 30)
- Secretary of State Mike Pompeo (Sept. 27)